As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the theorem(mainly through successive approximations) proves that the solution exists and is unique ONLY in that particular region. I believe that there exists a theorem which proves, that the Picard theorem implies existence and uniqueness for the whole Interval that the given function is defined on. However I can not for the life of me find that theorem. So how does this work? Am I correct in what I stated above? If I am, then why isn't that theorem as easy to find as Picard's theorem? It would seem that it's just as important. Could someone explain the theorem and perhaps give or help me find a proof for it?